Abstract

We prove that the Chern-Simons-Schrödinger system, under the condition of a Coulomb gauge, has a unique local-in-time solution in the energy space . The Coulomb gauge provides elliptic features for gauge fields . The Koch- and Tzvetkov-type Strichartz estimate is applied with Hardy-Littlewood-Sobolev and Wente's inequalities.

1. Introduction

We study herein the initial value problem of the Chern-Simons-Schrödinger (CSS) equations
where denotes the imaginary unit; , , and for ; is the complex scalar field; is the gauge field; is the covariant derivative for , and is a coupling constant representing the strength of interaction potential. The summation convention used involves summing over repeated indices and Latin indices are used to denote .

The CSS system of equations was proposed in [1, 2] to deal with the electromagnetic phenomena in planar domains, such as the fractional quantum Hall effect or high-temperature superconductivity. We refer the reader to [3, 4] for more information on the physical nature of these phenomena.

The CSS system exhibits conservation of mass
and the conservation of total energy
Note that the terms are missing in (3) when compared to the Maxwell-Schrödinger equations studied in [5].

To figure out the optimal regularity for the CSS system, we observe that the CSS system is invariant under scaling:
Therefore, the scaled critical Sobolev exponent is for . In view of (2) we may say that the initial value problem of the CSS system is mass critical.

The CSS system is invariant under the following gauge transformations:
where is a smooth function. Therefore, a solution to the CSS system is formed by a class of gauge equivalent pairs . In this work, we fix the gauge by imposing the Coulomb gauge condition of , under which the Cauchy problem of the CSS system may be reformulated as follows:
where the initial data . For the formulation of (6)–(8) we refer the reader to Section 3.

The initial value problem of the CSS system was investigated in [6, 7]. It was shown in [6] that the Cauchy problem is locally well posed in , and that there exists at least one global solution, , provided that the initial data are made sufficiently small in by finding regularized equations. They also showed, by deriving a virial identity, that solutions blow up in finite time under certain conditions. Explicit blow-up solutions were constructed in [8] through the use of a pseudo-conformal transformation. The existence of a standing wave solution to the CSS system has also been proved in [9, 10].

The adiabatic approximation of the Chern-Simons-Schrödinger system with a topological boundary condition was studied in [11], which provides a rigorous description of slow vortex dynamics in the near self-dual limit.

Taking the conservation of energy (3) into account, it seems natural to consider the Cauchy problem of the CSS system with initial data . Our purpose here is to supplement the original result of [6] by showing that there is a unique local- in-time solution in the energy space . We follow a rather direct means of constructing the solution and prove the uniqueness. We adapt the idea discussed in [12, 13] where a low regularity solution of the modified Schrödinger map (MSM) was studied. In fact, the CSS and MSM systems have several similarities except for the defining equation for . In the MSM, can be written roughly as , where denotes the Riesz transform. The local existence of a solution to the MSM was proved in [12] for the initial data in with , and similarly, the uniqueness was proved in [14] for with . To show the existence and uniqueness of the solution to the CSS system, the estimate of the gauge field, , is important for situations in which special structures of nonlinear terms in the defining equation for are used. The following describes are our main results.

Theorem 2. Let and be solutions to (6)–(8) on in the distribution sense with the same initial data to that outlined vide supra. Moreover, one assumes that
for some constant . One then has for .

We present some preliminaries in Section 2. Theorems 1 and 2 are proved in Sections 3 and 4, respectively. We conclude the current section by providing a few notations. We denote space time derivatives by and is used for spacial derivatives. We use the standard Sobolev spaces , with the norm and with the norm , where and . The space denotes . We define the space time norm as . We use to denote various constants. Because we are interested in local solutions, we may assume that . Thus, we replace the smooth function of with . We also use the convention of writing as shorthand for .

2. Preliminaries

We collect here a few lemmas used for the proof of Theorems 1 and 2. The following lemma is reminiscent of Wente's inequality (see [15, 16]).

Lemma 3. Let and be two functions in and let be the solution of
where is small at infinity. Then, and

The following energy estimate in [17, 18] is used for estimating a solution to the magnetic Schrödinger equation.

Lemma 4. Let u be a solution of
where and are real-valued functions. Then, for there exists an absolute constant such that
wherein one means the homogeneous Sobolev space when and simply when .

The following type of Strichartz estimate was used in [19, 20] for the study of the Benjamin-Ono equation. We refer to [12] for the counterpart to the Schrödinger equation.

Lemma 5. Let and be a solution to the equation
Then, for and , one has
where and .

We use the following Gagliardo-Nirenberg inequality with the specific constant [21], especially for the proof of Theorem 2.

Theorem 1 is proved in this section. Because the local well-posedness for smooth data is already known in [6], we simply present an a priori estimate for the solution to (6)–(8). Let us first explain (8). To derive it, note the following identities:
where and . Note that the second-order terms are cancelled out. Combined with the above algebra, the equation for comes from the second and third equations in (1):
We then have the formulation (6)–(8) in which is the only dynamical variable and , , and are determined through (7) and (8).

The constraint equation and the Coulomb gauge condition provide an elliptic feature of ; that is, the components can be determined from by solving the elliptic equations
Taking into account that the Coulomb gauge condition in Maxwell dynamics deduces a wave equation, the previous observation was used in [6]. Using (20), we have the following representation of :

3.1. Estimates for and

We are now ready to estimate several quantities of . Making use of (20) and the representation (21), we obtain the following estimates for .

Proposition 7. Let and . One also assumes that if or if . Then, one has

Proof. The above can be checked by applying Calderon-Zygmund and Hardy-Littlewood-Sobolev inequalities. We refer to [2, Section 2] for the details.

To estimate , the special algebraic structure and divergence form of the nonlinear terms in (19) are used.

Proof. Decompose as follows:
We first estimate the quantity . Applying Lemma 3 to (24), we deduce that
To estimate we use the Gagliardo-Nirenberg inequality with small :
Applying Hardy-Littlewood-Sobolev's inequality to (25) we deduce
where Proposition 7 and Lemma 6 are used. We can also derive the following from (25):
The first term can be estimated as follows:
where is used. The second term can be estimated as follows:
where is used. Therefore, we obtain with , that is, ,
Therefore, we conclude that
On the other hand, Lemma 3 shows that
We also have from (25) that
Therefore, we have

3.2. The Energy Solution to (CSS)

We now prove Theorem 1. Let us define
where , , and . We derive the following estimate:
from which Theorem 1 is proved by standard argument; see [2, Section 3].

Proposition 9. Let be a solution to (6)–(8). Then, one has
where and .

Proof. From the conservation of mass, we derive the first estimate. We apply Lemma 4 to (6) with and . Combined with Proposition 7, we have
where . We are then left to estimate . By Proposition 8, we obtain
Combining (40) and (41), we obtain
where and .

Proposition 10. Let be a solution to (6)–(8). Then, one has
where , and .

Proof. Applying Lemma 5 with and , we obtain
where , and . Considering Proposition 8, we obtain
The other terms can be treated, as mentioned in Section 1, by similar arguments to those in [2, Section 3]. Applying Proposition 7, we have
Plugging estimates (45)–(48) into (44) with , we obtain

We finally obtain the estimate (38) by combining Propositions 9 and 10, which proves Theorem 1.

In this section, we prove the uniqueness of the solution to (6). The basic rationale is borrowed from [12, 22].

Let and be solutions of (6)–(8) with the same initial data. If we set , then the equation for is
We will derive
where is a constant in Theorem 2 and . Then we have
Considering and , we obtain
Letting , for the time interval satisfying , we conclude that for , which thus proves Theorem 2.

In the remainder of this section, we derive inequality (51). Multiplying to both sides of (50) and integrating the imaginary part of , we have
The integrals (II)–(V), that is, those not containing , can be controlled by applying similar arguments to those described in [2, Section 4]. Integral (II) can be estimated, considering , by
for which we omit the proof.

We simply present how to control integral (I), for which we have
where , . Applying Lemma 6, we obtain
To control , we consider the equation for
Decomposing and as (24) and (25), we have
Taking into account
we can rewrite the equation for as follows:
where should be noted. Using the Hardy-Littlewood-Sobolev inequality, we have
where and , from which we deduce . Then, we have

The term can be bounded as follows:
Since , we have
Since , we may check
Then, we have
Combining estimates (57) and (69), and denoting , we obtain
where . We then obtain (51) by combining (55) and (70).

Acknowledgments

This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF), funded by the Ministry of Education, Science and Technology (2011-0015866), and was also partially supported by the TJ Park Junior Faculty Fellowship.